Quadratic forms classify products on quotient ring spectra

TL;DR

This paper explores the classification of products on quotient ring spectra using quadratic forms, revealing how bilinear and quadratic forms act on these products and their relation to commutativity.

Contribution

It introduces a group action of bilinear and quadratic forms on R-products of quotient spectra, linking algebraic forms to spectral product structures.

Findings

Group of bilinear forms acts freely and transitively on R-products.

Quadratic forms classify equivalence classes of R-products.

Characteristic bilinear form obstructs commutativity.

Abstract

We construct a free and transitive action of the group of bilinear forms Bil(I/I^2[1]) on the set of R-products on F, a regular quotient of an E-infinity ring spectrum R with F_* \cong R_*/I. We show that this action induces a free and transitive action of the group of quadratic forms QF(I/I^2[1]) on the set of equivalence classes of R-products on F. The characteristic bilinear form of F introduced by the authors in a previous paper is the natural obstruction to commutativity of F. We discuss the examples of the Morava K-theories K(n) and the 2-periodic Morava K-theories K_n.